Sum of Ideals is Ideal/Corollary

Corollary to Sum of Ideals is Ideal

Let $J_1$ and $J_2$ be ideals of a ring $\struct {R, +, \circ}$.

Let $J = J_1 + J_2$ be an ideal of $R$ where $J_1 + J_2$ is as defined in subset product.

Then:

$J_1 \subseteq J_1 + J_2$

$J_2 \subseteq J_1 + J_2$

Proof

From Sum of Ideals is Ideal we have that $j$ is an ideal of $R$.

Then:

$0_R \in J_1 + J_2$

and so:

$\forall x \in J_1: x + 0_R = x \in J_1 + J_2$

Similarly for $J_2$.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $40$